Exactly Solvable SOS Models II: Proof of the star-triangle relation and combinatorial identities

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

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Interaction between Different Kinds of Quantum Phase Transitions

Quantum Reports ◽

10.3390/quantum3020015 ◽

2021 ◽

Vol 3 (2) ◽

pp. 253-261

Author(s):

Angel Ricardo Plastino ◽

Gustavo Luis Ferri ◽

Angelo Plastino

Keyword(s):

Phase Transitions ◽

Nuclear Physics ◽

Body Problem ◽

Quantum Phase Transitions ◽

Quantum Phase ◽

Exactly Solvable Models ◽

Solvable Models ◽

Many Body ◽

Pairing Interaction ◽

Exactly Solvable

We employ two different Lipkin-like, exactly solvable models so as to display features of the competition between different fermion–fermion quantum interactions (at finite temperatures). One of our two interactions mimics the pairing interaction responsible for superconductivity. The other interaction is a monopole one that resembles the so-called quadrupole one, much used in nuclear physics as a residual interaction. The pairing versus monopole effects here observed afford for some interesting insights into the intricacies of the quantum many body problem, in particular with regards to so-called quantum phase transitions (strictly, level crossings).

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Fully degenerate Bell polynomials associated with degenerate Poisson random variables

Open Mathematics ◽

10.1515/math-2021-0022 ◽

2021 ◽

Vol 19 (1) ◽

pp. 284-296

Author(s):

Hye Kyung Kim

Keyword(s):

Random Variables ◽

Random Variable ◽

Combinatorial Identities ◽

Bell Polynomials ◽

Poisson Random Variable ◽

Central Moments ◽

Special Polynomials ◽

Special Numbers And Polynomials ◽

New Type ◽

Two Parameters

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

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A general inverse matrix series relation and associated polynomials – II

Mathematica Slovaca ◽

10.1515/ms-2017-0469 ◽

2021 ◽

Vol 71 (2) ◽

pp. 301-316

Author(s):

Reshma Sanjhira

Keyword(s):

Matrix Polynomial ◽

Combinatorial Identities ◽

Inverse Matrix ◽

Matrix Polynomials ◽

Special Cases ◽

Series Representations ◽

The Matrix ◽

Associated Polynomials ◽

Matrix Pair ◽

Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

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